Nonsingular black hole model as a possible end product of gravitational collapse

In this paper we present a nonsingular black hole model as a possible end-product of gravitational collapse. The depicted spacetime which is type $[II,(II)]$, by Petrov classification, is an exact solution of the Einstein equations and contains two horizons. The equation of state $p_r(\rho )$ in the radial direction, is a well-behaved function of the density $\rho (r)$ and smoothly reproduces vacuumlike behavior near $r=0$ while tending to a polytrope at larger $r$, low $\rho$ values. The final equilibrium configuration comprises a de Sitter-like inner core surrounded by a family of 2-surfaces $\Sigma$ of matter fields with variable equation of state. The fields are all concentrated in the vicinity of the radial center $r=0$. The solution depicts a spacetime that is asymptotically Schwarzschild at large $r$, while it becomes de Sitter-like as $r\to 0$. Possible physical interpretations of the macro-state of the black hole interior in the model are offered. We find that the possible state admits two equally viable interpretations, namely, either a quintessential intermediary region or a phase transition in which a two-fluid system is in both dynamic and thermodynamic equilibrium. We estimate the ratio of pure matter present to the total energy and in both cases find it to be virtually the same, being $\sim 0.83$. Finally, the well-behaved dependence of the density and pressure on the radial coordinate provides some insight on dealing with the information loss paradox.

Figure 1

The quartic equation of state $p(\rho )$ with $m=2,n=1$. At A $\rho =0$, $p=0$ and $\frac{dp}{d\rho }=0$. The value of the parameter $\alpha =2.213(5)$ is chosen so that for the entire density parameter space $\rho \ge 0$ the sound speed is not superluminal. The point B giving the maximum slope corresponds to the maximum sound speed, $\frac{d^{2}p}{d{r}^2}=0$, chosen to correspond to the light speed, $\frac{dp}{d\rho }=c^2=1$. Point C is the critical point $({\rho }_c,p_c)$, and for $\rho >{\rho }_c$, then $p(\rho )$ is a decreasing function. At point D or $\rho ={\rho }_{\epsilon }$, the pressure temporarily vanishes before it becomes negative.